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Theorem latleeqm1 14284
Description: Less-than-or-equal-to in terms of meet. (df-ss 3242 analog.) (Contributed by NM, 7-Nov-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latleeqm1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )

Proof of Theorem latleeqm1
StepHypRef Expression
1 latmle.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 latmle.l . . . . . . 7  |-  .<_  =  ( le `  K )
31, 2latref 14258 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
433adant3 975 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
54biantrurd 494 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<_  X  /\  X  .<_  Y ) ) )
6 simp1 955 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
7 simp2 956 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 simp3 957 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
9 latmle.m . . . . . 6  |-  ./\  =  ( meet `  K )
101, 2, 9latlem12 14283 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( X  .<_  X  /\  X  .<_  Y )  <->  X  .<_  ( X  ./\  Y )
) )
116, 7, 7, 8, 10syl13anc 1184 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  X  /\  X  .<_  Y )  <-> 
X  .<_  ( X  ./\  Y ) ) )
125, 11bitrd 244 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  X  .<_  ( X  ./\  Y )
) )
131, 2, 9latmle1 14281 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
1413biantrurd 494 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  ( X 
./\  Y )  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
1512, 14bitrd 244 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
16 latpos 14254 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
17163ad2ant1 976 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
181, 9latmcl 14256 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
191, 2posasymb 14185 . . 3  |-  ( ( K  e.  Poset  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X  ./\  Y ) )  <->  ( X  ./\ 
Y )  =  X ) )
2017, 18, 7, 19syl3anc 1182 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X 
./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) )  <-> 
( X  ./\  Y
)  =  X ) )
2115, 20bitrd 244 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   Posetcpo 14173   meetcmee 14178   Latclat 14250
This theorem is referenced by:  latleeqm2  14285  latnlemlt  14289  latabs2  14293  atnle  29576  2llnmat  29782  llnmlplnN  29797  dalem25  29956  2lnat  30042  lhpm0atN  30287  lhpmatb  30289  cdleme1  30485  cdleme5  30498  cdleme20d  30570  cdleme22e  30602  cdleme22eALTN  30603  cdleme23b  30608  cdleme32e  30703  doca2N  31385  djajN  31396  dihglblem5aN  31551  dihmeetbclemN  31563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-glb 14208  df-meet 14210  df-lat 14251
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